Recent progress in study of singular perturbation problems

Some results for the singular perturbation theory, methods and applications in the new century are reviewed in this paper. It could be found that in the recent decade, many approximate methods have been developed and refined, including the method of averaging, boundary layer methods, methods of matched asymptotic expansion and methods of multiple scales. An overview is given on the new work about various problems, such as the reaction diffusion, turning points, boundary layers, shock waves, stability problem, solitons, attractors, canard solution, scattering light wave and neuron network, etc. And then, a great number of applied problems are solved in applied mathematics, computational mathematics, fluid mechanics, elastic mechanics, optics, thermophysics, quantum mechanics, plasm physics, physical chemistry, analytical chemistry, epidemiology, neurology, engineering science, environment science, bionomics, atmosphere physics, ocean climate, aeronautics and astrnatics and so on, from which only some examples are extracted and described in this review.     

Singular perturbation of a second-order three-point boundary value problem for nonlinear systems

This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- and upper-solution pair, a concept defined in this paper, and employ the Nagumo conditions and algebraic boundary layer functions to ensure the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems have nonlinear dependence on all order derivatives of the unknown.     

一类高次非线性奇摄动问题的匹配解法

在适当的条件下,利用奇摄动理论中的收缩变换以及匹配技巧,讨论了一类高次非线性奇摄动两点边值问题.相应于边界条件.构造了原同题的外部解和内部解.按ε的增幂展开非线性项、比较同次幂的系数.再利用特异极限,由匹配原问题的内、外部解得出了相应于参数k=0、k=1、k<0、k>1和0相似文献   

基于过半可信原则的三I算法模糊控制器及其响应能力

给出了49个模糊蕴涵算子基于过半可信原则的三I算法的表达式,在此基础上详细讨论这49个模糊蕴涵算子基于过半可信原则三I算法构成的单输入单输出和双输入单输出模糊控制器,并分析了它们的响应能力．结果表明：在这49个模糊蕴涵算子中,只有一个模糊蕴涵箅子基于过半可信原则三I算法构成的模糊控制器具有函数的泛逼近性;另有一个模糊蕴涵算子基于过半可信原则三I算法构成的模糊控制器近似为拟合函数;44个模糊蕴涵算子基于过半可信原则三I算法构成的模糊控制器只具有阶跃响应能力;3个模糊蕴涵算子基于过半可信原则三I算法构成的模糊控制器为未定型．     

一类矩阵方程组的双对称解及其最佳逼近

建立了求矩阵方程组的双对称解的迭代算法.使用该方法不仅可以判断矩阵方程组是否有双对称解,而且在有双对称解时,还能够在有限步迭代计算之后得到矩阵方程组的双对称极小范数解.同时,也能够在矩阵方程组的对称解集合中求得给定矩阵的最佳逼近.     

Least-squares function-valued Padé-type approximation and its application in solving integral equations

To solve the Fredholm integral equations of the second kind, a new notion of the least-squares orthogonal polyno- mials of function-valued Pade-type approximation is introduced. On the basis of the error formula, the least-squares function： valued Pad＆type approximation is constructed. Their existence and uniqueness are studied. A recursive computation formula of the least-squares function-valued Padetype approximation is given. In the end, an example is given to show that the method is effective and stable.     

氢原子第四激发态（n=5）能级简并的解除

用简并微扰论研究氢原子第四激发态（n=5）在正交非均匀外电场中的能级分裂情况.并介绍高阶方阵本征问题的Jacobi解法.     

一类周期边值问题解的存在性及其奇异摄动

首先利用微分不等式研究如下类型的周期边值问题{y＇＇=f（t,y,y＇）,0〈t〈l y（0）=y（t）,y＇（0）=y＇（l）.的微分不等式与解的存在性,并在一定条件下把解延拓成为微分方程在无穷时间区N-Y-N期解,然后利用所得结果讨论奇异摄动现象．     

矩阵方程组异类约束解的MCG1-3-5算法

借鉴求线性矩阵方程组同类约束解的MCG算法（修正共轭梯度法）,建立了求多个未知矩阵的线性矩阵方程组的一种异类约束解的MCG1-3-5算法,证明了该算法的收敛性。该算法不仅可以判断矩阵方程组的异类约束解是否存在,而且在有异类约束解,且不考虑舍入误差时,可在有限步计算后求得矩阵方程组的一组异类约束解;选取特殊初始矩阵时,求得矩阵方程组的极小范数异类约束解。同时还能求取指定矩阵在该矩阵方程组异类约束解集合中的最佳逼近。算例表明,该算法有效。     

一维装箱问题的交叉算法分析

2004年孙春玲等研究了一维装箱问题,给出了一个近似程度最好的近似值为3/2的近似算法-交叉算法.遗憾的是他们的交叉算法的近似值分析是错误的,本文通过两个反例说明了他们的错误所在,并给出一个正确的近似值分析.